ALGEBRA

PROBLEM 8.

From Kepler's third law, we see that the farther a planet is located from the Sun, the longer its period is. Suppose Earth orbits the Sun in a circle of radius r (r = 1.5 x 10^8 km) with a period T (T = 1 year). Then any spacecraft SC (see Fig. 3.2) orbiting the Sun in the same plane but at some greater distance (r + a) will have a period larger than T, and if it starts from a point on the extension of the Earth-Sun line (as shown), it will gradually lag farther and farther behind.

However, the situation changes if a is sufficiently small, because then the gravity of Earth, in the configuration shown, adds appreciably to that of the Sun. For the force holding the spacecraft in orbit to balance the combined pull of the Earth and Sun, the spacecraft must move a bit faster. In fact, there is a particular value for a so that the speeding up of the spacecraft is just sufficient to allow it to keep up with Earth. If that happens, then the spacecraft orbits the Sun in a circle of radius (r + a), but with period T like Earth. What is the value of a that allows such an orbit?

The position we have found in this problem is an equilibrium point of the Sun-Earth system. A similar analysis can be used to show the existence of another equilibrium point on the sunward side of Earth, and in fact there are five such equilibrium points in all for any two-body gravitational system. These are called Lagrangian points in honor of the mathematician who first proposed their existence (Fig. 3.3). It has been suggested that two of the Lagrangian points of the Earth-Moon system should be considered as possible locations for future space colonies.

The ISEE-3 satellite (third International Sun-Earth Explorer), a joint venture of NASA and the European Space Agency, was launched in August 1978 and placed in a "halo orbit" around the Lagrangian equilibrium point between Earth and the Sun. In this orbit, it monitored the Sun's emissions that approach Earth‹without the interference that would result if the satellite were actually at the equilibrium point where its radio antenna would have to point directly at the Sun. After this mission was successfully completed in 1982, ISEE-3's orbit and direction were changed to conduct an exploratory survey of Earth's magnetotail. In December 1983, the satellite was redirected toward the comet Giacobini-Zinner and renamed International Cometary Explorer (ICE) in keeping with its new mission. It reached this comet in September 1985.

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